Twistor spaces of hypercomplex manifolds are balanced |
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Authors: | Artour Tomberg |
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Affiliation: | 1. Faculty of Mathematics, National Research University Higher School of Economics, 7 Vavilova Str., Moscow, 117312, Russia;2. Department of Mathematics and Statistics, McGill University, 805 Sherbrooke Street West, Montreal, Quebec, H3A 0B9, Canada |
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Abstract: | A hypercomplex structure on a differentiable manifold consists of three integrable almost complex structures that satisfy quaternionic relations. If, in addition, there exists a metric on the manifold which is Hermitian with respect to the three structures, and such that the corresponding Hermitian forms are closed, the manifold is said to be hyperkähler. In the paper “Non-Hermitian Yang–Mills connections” [13], Kaledin and Verbitsky proved that the twistor space of a hyperkähler manifold admits a balanced metric; these were first studied in the article “On the existence of special metrics in complex geometry” [17] by Michelsohn. In the present article, we review the proof of this result and then generalize it and show that twistor spaces of general compact hypercomplex manifolds are balanced. |
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Keywords: | Hypercomplex geometry Twistor theory Balanced manifold |
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